forked from Khan/khan-exercises
/
domain_of_a_function.html
275 lines (217 loc) · 18.4 KB
/
domain_of_a_function.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
<!DOCTYPE html>
<html data-require="math math-format">
<head>
<meta charset="UTF-8" />
<title>Domain of a function</title>
<script src="../khan-exercise.js"></script>
<script type="text/javascript">
function filter( o, f ) {
var n = {};
for ( var p in o ) {
if ( o.hasOwnProperty( p ) && f( p ) ) n[p] = o[p];
}
return n;
}
</script>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="A">randRange( 1, 10 )</var>
<var id="B" data-ensure="A !== B">randRange( 1, 10 )</var>
<var id="C" data-ensure="A !== C && B !== C">randRange( 1, 10 )</var>
<var id="SET">function() { return "\\{ \\, x \\in \\RR \\mid " + Array.prototype.join.call( arguments, ", \\," ) + "\\, \\}"; }</var>
<var id="FN">function( n, sym ) { return "x " + sym + n; }</var>
<var id="FN2">function( n, m, sym1, sym2 ) { return n + sym1 + " x " + sym2 + m; }</var>
<var id="NEQ">function( n ) { return FN( n, "\\neq" ); }</var>
<var id="GEQ">function( n ) { return FN( n, "\\geq" ); }</var>
<var id="LEQ">function( n ) { return FN( n, "\\leq" ); }</var>
<var id="GE">function( n ) { return FN( n, ">" ); }</var>
<var id="LE">function( n ) { return FN( n, "<" ); }</var>
<var id="LE_LEQ">function( n, m ) { return FN2( n, m, "<", "\\leq" ); }</var>
<var id="LEQ_LEQ">function( n, m ) { return FN2( n, m, "\\leq", "\\leq" ); }</var>
<var id="LE_LE">function( n, m ) { return FN2( n, m, "<", "<" ); }</var>
<var id="CHOICES">{
"two-denom-simplify": SET( NEQ( -1*A ), NEQ( B ) ),
"two-denom-cond": SET( NEQ( -1*A ) ),
"sqrt": SET( GEQ( A ) ),
"inverse-sqrt": SET( GE( A ) ),
"inverse-sqrt-cond": SET( NEQ( A ) ),
"sqrt-frac": SET( LE_LEQ( A, A+B ) ),
"two-denom-cond-weird": SET( NEQ( -1*A ), NEQ( C ) ),
"sqrt-poly-frac": SET( GEQ( C ) ),
"sqrt-abs": SET( LEQ_LEQ( -1*A, A ) ),
"inverse-sqrt-abs": SET( LE_LE( -1*A, A ) )
}</var>
</div>
<div class="problems">
<div id="two-denom-simplify">
<p class="problem"><code>f(x) = \dfrac{ x + <var>A</var> }{ ( x + <var>A</var> )( x - <var>B</var> ) }</code></p>
<p class="question">What is the domain of <code>f(x)</code>?</p>
<p class="solution"><code><var>CHOICES["two-denom-simplify"]</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<li data-each="filter( CHOICES, function( c ) { return c !== 'two-denom-simplify'; } ) as c"><code><var>c</var></code></li>
</ul>
<div class="hints">
<p><code>f(x)</code> is undefined when the denominator is 0.</p>
<p>The denominator is 0 when <code>x=<var>(-1*A)</var></code> or <code>x=<var>B</var></code>.</p>
<p>So we know that <code>x \neq <var>-1*A</var></code> and <code>x \neq <var>B</var></code>.</p>
<p>Expressing this mathematically, the domain is <code><var>CHOICES["two-denom-simplify"]</var></code>.</p>
</div>
</div>
<div id="two-denom-cond" data-type="two-denom-simplify">
<p class="problem"><code>f(x)= \begin{cases} \dfrac{ x + <var>A</var> }{ ( x + <var>A</var> )( x - <var>B</var> ) } & \text{if $x \neq <var>B</var>$} \\ <var>C</var> & \text{if $x = <var>B</var>$} \end{cases}</code></p>
<p class="solution"><code><var>CHOICES["two-denom-cond"]</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<li data-each="filter( CHOICES, function( c ) { return c !== 'two-denom-cond'; } ) as c"><code><var>c</var></code></li>
</ul>
<div class="hints">
<p><code>f(x)</code> is a piecewise function, so we need to examine where each piece is undefined.</p>
<p>The first piecewise definition of <code>f(x)</code>, <code>\frac{ x + <var>A</var> }{ ( x + <var>A</var> )( x - <var>B</var> ) }</code>, is undefined when its denominator is 0.</p>
<p>The denominator is 0 when <code>x=<var>-1*A</var></code> or <code>x=<var>B</var></code>.</p>
<p>So, based on the first piecewise definition, we know that <code>x \neq <var>-1*A</var></code> and <code>x \neq <var>B</var></code>.</p>
<p>However, the second piecewise definition applies when <code>x = <var>B</var></code>, and the second piecewise definition, <code><var>C</var></code>, has no weird gaps or holes, so <code>f(x)</code> is defined at <code>x = <var>B</var></code>.</p>
<p>So the only restriction on the domain is that <code>x \neq <var>-1*A</var></code>.</p>
<p>Expressing this mathematically, the domain is <code><var>CHOICES["two-denom-cond"]</var></code>.</p>
</div>
</div>
<div id="sqrt" data-type="two-denom-simplify">
<p class="problem"><code>f(x) = \sqrt{ x - <var>A</var> }</code></p>
<p class="solution"><code><var>CHOICES["sqrt"]</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<li data-each="filter( CHOICES, function( c ) { return c !== 'sqrt'; } ) as c"><code><var>c</var></code></li>
</ul>
<div class="hints">
<p><code>f(x)</code> is undefined when the radicand (the expression under the radical) is less than zero.</p>
<p>So the radicand, <code>x - <var>A</var></code>, must be greater than or equal to zero.</p>
<p>So <code>x - <var>A</var> \geq 0</code>; this means <code>x \geq <var>A</var></code>.</p>
<p>Expressing this mathematically, the domain is <code><var>CHOICES["sqrt"]</var></code>.</p>
</div>
</div>
<div id="inverse-sqrt" data-type="two-denom-simplify">
<p class="problem"><code>f(x) = \dfrac{ 1 }{ \sqrt{ x - <var>A</var> } }</code></p>
<p class="solution"><code><var>CHOICES["inverse-sqrt"]</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<li data-each="filter( CHOICES, function( c ) { return c !== 'inverse-sqrt'; } ) as c"><code><var>c</var></code></li>
</ul>
<div class="hints">
<p>First, we need to consider that <code>f(x)</code> is undefined when the radicand (the expression under the radical) is less than zero.</p>
<p>So the radicand, <code>x - <var>A</var></code>, must be greater than or equal to zero.</p>
<p>So <code>x - <var>A</var> \geq 0</code>; this means <code>x \geq <var>A</var></code>.</p>
<p>Next, we also need to consider that <code>f(x)</code> is undefined when the denominator, <code>\sqrt{ x - <var>A</var> }</code>, is zero.</p>
<p>So <code>\sqrt{ x - <var>A</var> } \neq 0</code>.</p>
<p><code>\sqrt{ z } = 0</code> only when <code>z = 0</code>, so <code>\sqrt{ x - <var>A</var> } \neq 0</code> means that <code>x - <var>A</var> \neq 0</code>.</p>
<p>So <code>x \neq <var>A</var></code>.</p>
<p>So we have two restrictions: <code>x \geq <var>A</var></code> and <code>x \neq <var>A</var></code>.</p>
<p>Combining these two restrictions, we are left with simply <code>x > <var>A</var></code>.</p>
<p>Expressing this in mathematical notation, the domain is <code><var>CHOICES["inverse-sqrt"]</var></code>.</p>
</div>
</div>
<div id="inverse-sqrt-cond" data-type="two-denom-simplify">
<p class="problem"><code>f(x) = \begin{cases} \dfrac{ 1 }{ \sqrt{ x - <var>A</var> } } & \text{if $x \geq <var>A</var>$} \\ \dfrac{ 1 }{ \sqrt{ <var>A</var> - x } } & \text{if $x < <var>A</var>$} \end{cases}</code></p>
<p class="solution"><code><var>CHOICES["inverse-sqrt-cond"]</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<li data-each="filter( CHOICES, function( c ) { return c !== 'inverse-sqrt-cond'; } ) as c"><code><var>c</var></code></li>
</ul>
<div class="hints">
<p><code>f(x)</code> is a piecewise function, so we need to examine where each piece is undefined.</p>
<p>The first piecewise definition of <code>f(x)</code>, <code>\frac{ 1 }{ \sqrt{ x - <var>A</var> } }</code>, is undefined where the denominator is zero and where the radicand (the expression under the radical) is less than zero.</p>
<p>The denominator, <code>\sqrt{ x - <var>A</var> }</code>, is zero when <code>x - <var>A</var> = 0</code>, so we know that <code>x \neq <var>A</var></code>.</p>
<p>The radicand, <code>x - <var>A</var></code>, is less than zero when <code>x < <var>A</var></code>, so we know that <code>x \geq <var>A</var></code>.</p>
<p>So the first piecewise definition of <code>f(x)</code> is defined when <code>x \neq <var>A</var></code> and <code>x \geq <var>A</var></code>. Combining these two restrictions, the first piecewise definition is defined when <code>x > <var>A</var></code>. The first piecewise defintion applies when <code>x \geq <var>A</var></code>, so this restriction is relevant.</p>
<p>The second piecewise definition of <code>f(x)</code>, <code>\frac{ 1 }{ \sqrt{ <var>A</var> - x } }</code>, applies when <code>x < <var>A</var></code> and is undefined where the denominator is zero and where the radicand is less than zero.</p>
<p>The denominator, <code>\sqrt{ <var>A</var> - x }</code>, is zero when <code><var>A</var> - x = 0</code>, so we know that <code>x \neq <var>A</var></code>.</p>
<p>The radicand, <code><var>A</var> - x</code>, is less than zero when <code>x > <var>A</var></code>, so we know that <code>x \leq <var>A</var></code>.</p>
<p>So the second piecewise definition of <code>f(x)</code> is defined when <code>x \neq <var>A</var></code> and <code>x \leq <var>A</var></code>. Combining these two restrictions, the second piecewise definition is defined when <code>x < <var>A</var></code>. However, the second piecewise definition of <code>f(x)</code> only applies when <code>x < <var>A</var></code>, so restriction isn't actually relevant to the domain of <code>f(x)</code>.</p>
<p>So the first piecewise definition is defined when <code>x > <var>A</var></code> and applies when <code>x \geq <var>A</var></code>; the second piecewise definition is defined when <code>x < <var>A</var></code> and applies when <code>x < <var>A</var></code>. Putting the restrictions of these two together, the only place where a definition applies and the value is undefined is at <code>x = <var>A</var></code>. So the only restriction on the domain of <code>f(x)</code> is <code>x \neq <var>A</var></code>.</p>
<p>Expressing this mathematically, the domain is <code><var>CHOICES["inverse-sqrt-cond"]</var></code>.</p>
</div>
</div>
<div id="sqrt-frac" data-type="two-denom-simplify">
<p class="problem"><code>f(x) = \dfrac{ \sqrt{ <var>A+B</var> - x } }{ \sqrt{ x - <var>A</var> } }</code></p>
<p class="solution"><code><var>CHOICES["sqrt-frac"]</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<li data-each="filter( CHOICES, function( c ) { return c !== 'sqrt-frac'; } ) as c"><code><var>c</var></code></li>
</ul>
<div class="hints">
<p>First, we need to consider that <code>f(x)</code> is undefined anywhere where either radical is undefined, so anywhere where either radicand (the expression under the radical symbol) is less than zero.</p>
<p>The top radical is undefined when <code><var>A+B</var> - x < 0</code>.</p>
<p>So the top radical is undefined when <code>x > <var>A+B</var></code>, so we know <code>x \leq <var>A+B</var></code>.</p>
<p>The bottom radical is undefined when <code>x - <var>A</var> < 0</code>.</p>
<p>So the bottom radical is undefined when <code>x < <var>A</var></code>, so we know <code>x \geq <var>A</var></code>.</p>
<p>Next, we need to consider that <code>f(x)</code> is undefined anywhere where the denominator, <code>\sqrt{ x - <var>A</var> }</code>, is zero.</p>
<p>So <code>\sqrt{ x - <var>A</var> } \neq 0</code>, so <code>x - <var>A</var> \neq 0</code>, so <code>x \neq <var>A</var></code>.</p>
<p>So we have three restrictions: <code>x \leq <var>A+B</var></code>, <code>x \geq <var>A</var></code>, and <code>x \neq <var>A</var></code>.</p>
<p>Combining these three restrictions, we know that <code><var>A</var> < x \leq <var>A+B</var></code>.</p>
<p>Expressing this mathematically, the domain is <code><var>CHOICES["sqrt-frac"]</var></code>.</p>
</div>
</div>
<div id="two-denom-cond-weird" data-type="two-denom-simplify">
<p class="problem"><code>f(x) = \begin{cases} \dfrac{ x + <var>A</var> }{ ( x + <var>A</var> )( x - <var>C</var> ) } & \text{if $x \neq <var>B</var>$} \\ <var>A</var> & \text{if $x = <var>B</var>$} \end{cases}</code></p>
<p class="solution"><code><var>CHOICES["two-denom-cond-weird"]</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<li data-each="filter( CHOICES, function( c ) { return c !== 'two-denom-cond-weird'; } ) as c"><code><var>c</var></code></li>
</ul>
<div class="hints">
<p><code>f(x)</code> is a piecewise function, so we need to examine where each piece is undefined.</p>
<p>The first piecewise definition of <code>f(x)</code>, <code>\frac{ x + <var>A</var> }{ ( x + <var>A</var> )( x - <var>C</var> ) }</code>, is undefined where the denominator is zero.</p>
<p>The denominator, <code>(x + <var>A</var>)(x - <var>C</var>)</code>, is zero when <code>x = <var>-1*A</var></code> or <code>x = <var>C</var></code>.</p>
<p>So the first piecewise definition of <code>f(x)</code> is defined when <code>x \neq <var>-1*A</var></code> and <code>x \neq <var>C</var></code>. The first piecewise definition applies when <code>x = <var>-1*A</var></code> and <code>x = <var>C</var></code>, so these restrictions are relevant to the domain of <code>f(x)</code>.</p>
<p>The second piecewise definition of <code>f(x)</code>, <code><var>A</var></code>, is a simple horizontal line function and so has no holes or gaps to worry about, so it is defined everywhere.</p>
<p>So the first piecewise definition is defined when <code>x \neq <var>-1*A</var></code> and <code>x \neq <var>C</var></code> and applies when <code>x \neq <var>B</var></code>; the second piecewise definition is defined always and applies when <code>x = <var>B</var></code>. Putting the restrictions of these two together, the only places where a definition applies and is undefined are at <code>x = <var>-1*A</var></code> and <code>x = <var>C</var></code>. So the restriction on the domain of <code>f(x)</code> is that <code>x \neq <var>-1*A</var></code> and <code>x \neq <var>C</var></code>.</p>
<p>Expressing this mathematically, the domain is <code><var>CHOICES["two-denom-cond-weird"]</var></code>.</p>
</div>
</div>
<div id="sqrt-poly-frac" data-type="two-denom-simplify">
<p class="problem"><code>f(x) = \dfrac{ \sqrt{ x - <var>C</var> } }{ x^2 + <var>A+B</var> x + <var>A*B</var> }</code></p>
<p class="solution"><code><var>CHOICES["sqrt-poly-frac"]</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<li data-each="filter( CHOICES, function( c ) { return c !== 'sqrt-poly-frac'; } ) as c"><code><var>c</var></code></li>
</ul>
<div class="hints">
<p><code>f(x) = \dfrac{ \sqrt{ x - <var>C</var> } }{ x^2 + <var>A+B</var> x + <var>A*B</var> } = \dfrac{ \sqrt{ x - <var>C</var> } }{ ( x + <var>A</var> )( x + <var>B</var> ) }</code></p>
<p>First, we need to consider that <code>f(x)</code> is undefined anywhere where the radical is undefined, so the radicand (the expression under the radical) cannot be less than zero.</p>
<p>So <code>x - <var>C</var> \geq 0</code>, which means <code>x \geq <var>C</var></code>.</p>
<p>Next, we also need to consider that <code>f(x)</code> is undefined anywhere where the denominator is zero.</p>
<p>So <code>x \neq <var>-1*A</var></code> and <code>x \neq <var>-1*B</var></code>.</p>
<p>However, these last two restrictions are irrelevant since <code><var>C</var> > <var>-1*A</var></code> and <code><var>C</var> > <var>-1*B</var></code> and so <code>x \geq <var>C</var></code> will ensure that <code>x \neq <var>-1*A</var></code> and <code>x \neq <var>-1*B</var></code>.</p>
<p>Combining these restrictions, then, leaves us with simply <code>x \geq <var>C</var></code>.</p>
<p>Expressing this mathematically, the domain is <code><var>CHOICES["sqrt-poly-frac"]</var></code>.</p>
</div>
</div>
<div id="sqrt-abs" data-type="two-denom-simplify">
<p class="problem"><code>f(x) = \sqrt{ <var>A</var> - \lvert x \rvert }</code></p>
<p class="solution"><code><var>CHOICES["sqrt-abs"]</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<li data-each="filter( CHOICES, function( c ) { return c !== 'sqrt-abs'; } ) as c"><code><var>c</var></code></li>
</ul>
<div class="hints">
<p><code>f(x)</code> is undefined when the radicand (the expression under the radical) is less than zero.</p>
<p>So we know that <code><var>A</var> - \lvert x \rvert \geq 0</code>.</p>
<p>So <code>\lvert x \rvert \leq <var>A</var></code>.</p>
<p>This means <code>x \leq <var>A</var></code> and <code>x \geq <var>-1*A</var></code>; or, equivalently, <code><var>-1*A</var> \leq x \leq <var>A</var></code>.</p>
<p>Expressing this mathematically, the domain is <code><var>CHOICES["sqrt-abs"]</var></code>.</p>
</div>
</div>
<div id="inverse-sqrt-abs" data-type="two-denom-simplify">
<p class="problem"><code>f(x) = \dfrac{ <var>B</var> }{ \sqrt{ <var>A</var> - \lvert x \rvert } }</code></p>
<p class="solution"><code><var>CHOICES["inverse-sqrt-abs"]</var></code></p>
<ul class="choices" data-show="5" data-none="true">
<li data-each="filter( CHOICES, function( c ) { return c !== 'inverse-sqrt-abs'; } ) as c"><code><var>c</var></code></li>
</ul>
<div class="hints">
<p>First, we need to consider that <code>f(x)</code> is undefined anywhere where the radicand (the expression under the radical) is less than zero.</p>
<p>So we know that <code><var>A</var> - \lvert x \rvert \geq 0</code>.</p>
<p>This means <code>\lvert x \rvert \leq <var>A</var></code>, which means <code><var>-1*A</var> \leq x \leq <var>A</var></code>.</p>
<p>Next, we need to consider that <code>f(x)</code> is also undefined anywhere where the denominator is zero.</p>
<p>So we know that <code>\sqrt{ <var>A</var> - \lvert x \rvert } \neq 0</code>, so <code>\lvert x \rvert \neq <var>A</var></code>.</p>
<p>This means that <code>x \neq <var>A</var></code> and <code>x \neq <var>-1*A</var></code>.</p>
<p>So we have four restrictions: <code>x \geq <var>-1*A</var></code>, <code>x \leq <var>A</var></code>, <code>x \neq <var>-1*A</var></code>, and <code>x \neq <var>A</var></code>.</p>
<p>Combining these four, we know that <code>x > <var>-1*A</var></code> and <code>x < <var>A</var></code>; alternatively, that <code><var>-1*A</var> < x < <var>A</var></code>.</p>
<p>Expressing this mathematically, the domain is <code><var>CHOICES["inverse-sqrt-abs"]</var></code>.</p>
</div>
</div>
</div>
</div>
</body>
</html>